Questions tagged [self-study]

A routine exercise from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for such questions rather than complete answers.

2,056 questions with no upvoted or accepted answers
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8
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332 views

Sufficient statistics for $\mu_1 - \mu_2$

If $ X_1, ..., X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is known,...
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285 views

Rao-Blackwellization in variational inference

The Black box VI paper introduces Rao-Blackwellization as a method to reduce the variance of the gradient estimator using score function, in section 3.1. However I don't quite get the basic idea ...
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1k views

Exercise on Borel Cantelli Lemma ($\limsup X_n/ \ln(n) =1$ a.s.) help required to rigorously write the statement

I hope this question is within the scope of this site. Please note that I have solved this Exercise, I do have doubts about my presentation though and about how to rigorously empathize on the ...
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Question 10.9 from Bayesian Data Analysis, what does accuracy mean here?

I'm doing an independent study in Bayesian Statistics following some chapters from BDA3. When solving the first question from Ch 10 I got stuck. It says: [If] a scalar variable $\theta$ is ...
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79 views

Joint distribution of $Y$ and $S^2-Y^2$

Let $\{X_i\}_{i=1}^n\overset{iid}{\sim}\mathcal{N}(\mu,\sigma^2)$. Let $\{b_i\}_{i=1}^n$ be a sequence of numbers so that $\sum_{i=1}^nb_i=0$ and $\sum_{i=1}^nb^2_i=1$. Define $$S^2=\sum_{i=1}^n(X_i-\...
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203 views

Why does MLE tend to normal distribution

We have $X_1,\dots, X_n$ are iid (the distribution can be of any type, e.g. Bernoulli (p), normal ($\mu, \sigma^2$), Poisson ($\lambda$). If we use MLE $\hat \theta$ to estimate any parameter $\theta$ ...
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93 views

French website Providing Instruction/Tutorials on Statistical Theory

This is somewhat of an odd question for CV, but since it's a question about statistical education, I think it falls within the scope of CV. Several years ago I stumbled across a French website that ...
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220 views

Can an asymptotically efficient estimator be biased?

In "Theory of point estimation" by Lehmann and Casella (1998) there is the following definition: It is also said that So terms of the asymptotically normal sequence of estimators can be ...
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Dealing with auxiliary random variables for Mean-Field Variational Inference in Bayesian Poisson factorization

I am studying as a part of a class assignment a recent paper on Poisson factorization. Some points of the paper regarding the usage of some auxiliary variables are not clear to me. I would like to ...
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2k views

Deriving the maximum likelihood for a generative classification model for K classes

In Christopher Bishop's book "Pattern Recognition and Machine learning", there is the following question: Consider a generative classification model for $K$ classes defined by the prior class ...
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5k views

Conjugate of Weibull with shape known

This isn't exactly a homework problem but rather a self-selected problem I'm doing to prepare for a midterm. I can see from Wikipedia that it is an inverse gamma but I am unable to reach the ...
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154 views

Decision Theory: Why is it called a "least favorable prior"?

I'm currently reading the chapter on Statistical Decision Theory in Larry Wasserman's "All of Statistics". Reading the section 13.4 about Minimax Rules he introduces the so called Least ...
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1answer
184 views

Proving the nonexistence of UMVUE for $\text{Unif}\{\theta-1, \theta, \theta+1\}$

I am trying to prove that There is no UMVUE for $\theta$ for the distribution $\text{Unif}\{\theta-1, \theta, \theta+1\}$, $\theta$ is an integer. Here is what I have attempted. I am trying to use ...
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63 views

Asymptotic properties of functional models

When working in Functional Data Analysis, a classical "preprocessing" step is to represent the "observations" using a B-spline expansion: $$ X_i(t) \approx \sum_{j=1}^J \lambda_{ij}...
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73 views

How to show a UMVUE exists only if $g(p)$ is a polynomial of degree at most $n$?

Let $X\sim Bin(n,p)$. The problem is to show that a UMVUE can exist for $g(p)$ only if $g(p)$ is a polynomial in $p$ of degree at most $n$. For the case when $g(p) = \frac{1}{p}$ we can show that it ...
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708 views

Partitioned regression model: estimator of beta 1

below is an exercise that is really giving me a hard time, I believe that there is a simple way around it but I can not find it: Assume the correct regression model is Y = X$\beta$ + $\epsilon$ for E(...
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172 views

Does Cross-Validation Really Work?

This question was taken from "The Elements of Statistical Learning" by Friedman, Hastie and Tibshirani (question 7.10) Consider a scenario with N = 20 samples in two equal-sized classes, and p = 500 ...
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297 views

Variance of quotient of Poisson random variable and sum of the Poisson sample

Let $$Y_1\sim \operatorname{Poisson}(\lambda_1)\\Y_2\sim \operatorname{Poisson}(\lambda_2),$$ where $Y_1$ and $Y_2$ are independent, and $\lambda_1, \lambda_2>0$. What is the variance of $$\frac{...
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347 views

Describe AR process with additive white noise using ARMA process

Disclaimer: This is a homework problem This is a problem from "Adaptive Filter Theory" by Haykin. Problem 2.10 (2nd edition). Problem A discrete-time stochastic process $\{x(n)\}$ that is real-...
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I'm not asking for a conjugate prior. Is there a distribution $p(x|y)$ that satisfies $\int p(x|y)Beta(y|a,b) dy = Beta(x| a', b')$?

I know the result of integrating a Gaussian against another Gaussian is still Gaussian, $$\int N(x|\mu_y,\sigma_y)N(y|\mu,\sigma) dy = N(x|\mu',\sigma')\quad.$$ Can I get the same form for Beta ...
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1answer
209 views

Linear Regression - Confidence interval for mean response vs prediction interval

I understand the concept of a confidence interval for the mean response (fitted line) for simple linear regression $y$ = $\beta_{0}$+$\beta_{1}$$X_{i}$. It is that taken over many times, with 95% ...
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167 views

Derivation of the BIC

i am trying to self-study / understand the derivation of the BIC. I have studied that: however - it is not quite clear to me how this leads to the formula below. I don't fully understand where the ...
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696 views

Bayesian Analysis of Box-Cox Transformation

This problem is problem 5 in Chapter 7 of Bayesian Data Analysis, 3rd edition. Consider the Box-Cox transformation: $y_i^{(\lambda)} \sim \mathcal{N}(\mu, \sigma^2)$ where $y_i^{(\lambda)} = (y_i^{\...
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Combination of letters: some repeated letters

So I was looking for an answer on this assignment we have to program but I cannot find it anywhere. I'm a computer science student, not a statistics students. (And it isn't even for a statistics ...
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2k views

Neyman-Pearson lemma: critical region and hypothesis testing

Let $X_1,X_2,...,X_n$ be i.i.d r.v's with common p.d.f. $$ \mbox f(x)=\frac{x^5e^{-x/\theta}}{5!\theta^6} $$ where $\theta$ > 0. Show that the Neyman-Pearson lemma produces a test of $H_0: \...
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174 views

Simple $\chi^2$ test question

I have the following question which seems extremely easy, but the way the data are set up is causing me some uncertainty: I plan to solve this problem through finding the maximum likelihood estimate ...
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182 views

Exponential family where set of natural parameters has empty interior

In my math-stat class we have a theorem that goes: Let $\{P_\theta : \theta \in \Theta\}$ be a $k$ parameter exponential family (i.e. the density of a member of this family can be written as $f(\...
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319 views

Deriving priors for MCMC implementation

I have been working on an assignment lately wherein the object is to implement an MCMC approach to simulate from a generated posterior distribution. The posterior distribution is generated from a ...
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1answer
87 views

Efficient influence function in proportional hazards model

I was hoping someone could help me with this problem in the cox proportional hazards model. I am given the following setup. T is a non-negative random variable with continous distribution and hazard ...
4
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95 views

P-value of LR test

I've been studying more about GLRT (Generalized Likelihood Ratio Tests) and I came up with the following problem. Let $X\sim N(\theta,1)$ and consider the hypothesis $H_0:\theta\in[a,b]$ against $H_1:\...
4
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1answer
228 views

Finding the posterior mean

I have been trying to solve the following problem: Suppose $X_1,...,X_n$ are iid exponential random variables, with density $f(x;\theta) =\theta e^{-\theta x}$ ,and let us suppose that we have a prior ...
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94 views

Deriving spectral measure

While reading this book, I got stuck on page 266 where the authors found the spectral measure $F(du)$ of the generalized covariance function $K(h) = \Gamma(-\alpha/2) |h|^{\alpha}, ~0<\alpha<2.$ ...
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82 views

How is EMSE derived for causal trees in Athey and Imbens (PNAS 2016)?

Athey and Imbens build a non-parametric matching procedure to identify and estimate causal effects. To this end, they minimize the expected mean squared error (EMSE) of their procedure, but I don't ...
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51 views

Figuring out the margin for the soft margin SVM (exam question)

This is an exam question and I am not sure whether it is solveable with the given information. We were given a graphic that displayed binary labelled points $x^{(i)}\in \mathbb{R}^2$ with $y^{(i)} \...
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99 views

Clarification on the concept of a cumulative distribution function of a measure (measure theory)

I was asked to show that $g_f(x)=\mu(f\leq x)$ defines a cumulative distribution function for any measurable function $f$. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $(\mathbb{R},\...
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54 views

Integration by sampling from truncated distribution

I'm reading the book Ben Lambert's Bayesian Statistics: problems and answers, which by the way I like. There is a group of problems in "Integration by Sampling" chapter 12. The first integral is $$...
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232 views

Test for Lipschitz continuity (is there some?)

Let $x_1, \dots, x_n$ be a random sample from a distribution $D$. Say, I want to test whether $F(z)$, the cdf of $D$, is Lipschitz continuous, i.e. there exists $L$ such that $F(z + \delta) - F(z) \...
4
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247 views

Minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed

What is the minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed? median When we wish to estimate the median, $\mu$, of a normal distributed variable then ...
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52 views

Uniform distribution on the simplex. - Cover & Thomas

I'm trying to formulate the solution for the following problem: I was thinking in finding the equivalent distribution on $X_i$ based on $Y_i$, but I think I'm cheating. I think that the autor wants ...
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1answer
112 views

Convergence in Distribution for i.i.d. data

Let $X_1,X_2,\ldots,X_n$ be i.i.d. RVs with $E(X_{i})=\mu$ and $V(X_{i})=\sigma^2$, $\sigma <\infty$.Is it possible to find real sequences $a_{n}$ and $b_{n}$ such that $a_{n}(\bar{X}^3_{n}-b_{n})$ ...
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169 views

Show that there is no efficient estimator for the variance of a normal distribution using properties of the exponential family

I want to prove the statement in the title using the following statement from Wikipedia: it was proved that efficient estimation is possible only in an exponential family, and only for the natural ...
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101 views

Find unbiased estimators for $\lambda$ and $\lambda^2$.

For the spatial homogeneous Poisson process, find unbiased estimators for $\lambda$ and $\lambda^2$. Attempt: Since the homogeneous Poisson process is over an area, how i would i go about ...
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450 views

FDR and the the Benjamini-Hochberg Method

I am trying to understand the Benjamini-Hochberg Method for controlling the false discovery rate. Mathematically, if we are given with m hypothesis testing procedures, we sort the P-Values and reject ...
4
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1answer
400 views

What is the likelihood function of this random variable (beta distribution parameterizing a Bernoulli distribution)?

This is related to an earlier self-study question of mine. The setup is that there are $N$ individuals, indexed by $i$, and two time periods. Individuals choose whether to "invent" something in the ...
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1k views

Concentration of maximum of subexponential random variables

I'm looking for a concentration bound on the maximum of a collection of sub-exponential random variables, which are not necessarily independent. More specifically, I have the following collection: \...
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0answers
292 views

Voting between classifiers : How to prove it works?

Assume $m$ independent binary classifiers with probability $p$ to be correct $p>0.5$. Show that the probability of a voting, e.g. decision is made by the majority of classifiers is correct with ...
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612 views

Back-forecasting in MA(2) model

The sales of a certain product are represented by the model $$Z_t=3+a_t+0.5a_{t-1}-0.25a_{t-2}$$ where $a_t\sim WN(0,4)$ (White Noise). Given the data $Z_1=3.25,Z_2=4.75,Z_3=2.25$ and $Z_4=...
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383 views

Linear regression when the conditional distribution is Poisson

Suppose $Y$ is discrete and only takes on non-negative integers and that the conditional distribution of $Y$ given $X=x$ is Poisson, that is, $$P(Y=y|X=x) = \frac{\exp(-x'\beta) (x'\beta)^y}{y!}$...
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42 views

Can someone clearly paraphrase the following argument?

Metaculus is a site where users make and justify predictions on various questions. My question is about an estimation of the probability that a human will live to 120 years by the year 2024. I am ...
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406 views

Quiz: Determine first principal component from data-plots

We see four data plots. The goal: How does the first principal component look for each plot a-d. For plot d, it is true that both clusters have same number of datapoints. First principal component ...

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